## The Factor Game

• Lesson
6-8
3

The Factor Game engages students in a friendly contest in which winning strategies involve distinguishing between numbers with many factors and numbers with few factors. Students are then guided through an analysis of game strategies and introduced to the definitions of prime and composite numbers.

### Background Information

The Factor Game is a two-person game in which players find factors of numbers on a game board. To play, one person selects a number and colors it. The second person colors all the proper factors of the first person's number. The roles are switched and the play continues till there are no numbers remaining with uncolored factors. Each person adds up the numbers they've colored. The winner is the person with the largest total.

The purpose of this investigation is twofold: to help students determine whether a given number has many or only a few factors and to show how this property of numbers is useful for problem solving. Part I, Playing the Factor Game, engages students in a friendly contest in which winning strategies involve recognizing the difference between prime numbers and composite numbers. Part II, Playing to Win the Factor Game, guides students through an analysis of Factor Game strategies and introduces the definitions of prime and composite numbers. Part III provides questions that are rich in connections to situations in which factors, multiples, divisors, products, and prime numbers are significant.

This investigation is based on the Factor Game from the Prime Time unit of the Connected Mathematics Project, G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1996) pp.1-16.

### Conducting the Investigation

#### Day 1

Launch

This problem gives students an opportunity to learn about factors by playing a two-person board game. On each turn, one player chooses a number, and the other player finds the factors of that number. While playing the game, students become familiar with the factors of the numbers from 2 to 30 and review multiplication and division of small whole numbers.

Discuss the material at the top of page one and elaborate on what a factor of a number is. Remind students that there are two ways to think of a factor: as one of the numbers that is multiplied to get a product, and as a divisor of a number. You could ask students to give examples:

What factors can you multiply to get a product of 10?

When you feel your students understand what a factor is, introduce the Factor Game. The rules for the game and a sample game are given in the investigation, but the best way to get students started is to play a game against the class using the game board, in the Activity Sheets Packet. Rather than reading all the rules at the start, explain the rules as the need arises during the game. When you play against the class, we suggest that you take your turn first and that you choose a non-prime number, such as 26, for your first move. This way, students can discover the power of a prime first move. Students can either use printed copies of the game board or use the provided interactive applet.

A Note on Calculators

In the Connected Mathematics curriculum, we assume that students have access to calculators at all times. However, we hope that students will develop good estimation and mental arithmetic skills. This means that you need to give your students guidelines about the appropriate uses of calculators. In some classes, students may be ready to do all of the arithmetic in the Factor Game without the help of calculators. In other classes, students may need to use calculators to check their mental computations. You need to make a judgment call about whether to use the game as an opportunity for practice in mental arithmetic or to encourage your students to use calculators. After students have a sense of the Factor Game, you may find it appropriate to encourage them to use calculators to keep running totals of their scores.

Explore

Have students play the game two or three times with a partner.

As you play the game, think about the questions I am writing on the board.

Write the following questions on the board:

Is it better to go first or second? Why?

What is the best first move? Why?

How do you know when the game is over?

Summarize

You may want to have a few students share some ideas they discovered while playing the game. However, since the next part is an analysis of the game, you can delay an extensive summary until then.

#### Day 2

Launch

This problem engages students in systematically analyzing the Factor Game. The questions you wrote on the board in the Explore section of the last part help to launch this part.

Thinking about the best first move in the Factor Game makes me wonder what the results would be for each number if I chose it as my first move. What if I chose 1 or 2 or 3? How many points would I get? How many points would my opponent get?

Let's find the results for every possible first move. Can you think of a way that we can organize our work so that we can see patterns and determine which moves are good and which moves are bad?

Give your students a chance to suggest ways to approach the analysis and organization of their work. If no good ideas surface, have them consider tabular organization. Once the students and you have agreed on a scheme for organizing the data, remind them of what they are trying to determine. (If you want to provide your students with a chart for recording, see the handouts.)

Remember, you are exploring what your score and your opponent's score would be if you chose each of the numbers from 1 through 30 as your first move. When your chart is complete, write the answers for Part 1and Part 2 in your journal.

Explore

Give students 5 to 10 minutes to work on their charts individually. Then allow time for them to work with a partner to compare, correct, and complete their charts. You may want to make and display a class chart so you and your students can refer to it during the rest of the unit. A possible chart is given in the handouts.

A Note on Factor Game First Moves

The chart indicates that prime numbers are good first moves, especially large primes like 29. (Note that prime numbers are only legal when they are first moves. Once a first move has been made, all primes are illegal because their only proper factor, 1, will have already been circled.) This chart is also a good display of abundant, deficient, and perfect numbers (explained in Part 3). The number 24, for example, is abundant because the sum of its proper factors is more than 24. The number 16 is deficient because the sum of its proper factors is less than 16. The number 6 is perfect because the sum of its proper factors equals 6. Note that 6 and 28 are the only perfect numbers between 1 and 30.

Summarize

A good way to summarize is to have students share their answers to Part 1 and Part 2 with the class. Students should record the results of the discussion in their journals, either in class or as a part of their homework. You may wish to discuss the fact that the number 1 is neither prime nor composite because it has no proper factors at all.

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### Learning Objectives

Students will be able to:

• Classify numbers as prime or composite.
• Recognize that some numbers are rich in factors, while others have few factors.
• To recognize that factors come in pairs and that once one factor is found, another can also be found.
• To discover the connection between dividing and finding factors of a number.

### NCTM Standards and Expectations

• Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.
• Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.
• Develop and analyze algorithms for computing with fractions, decimals, and integers and develop flue